^ 





ON 



THE NEW FOEM 



THE ACHROMATIC OBJECT-GLASS 



INTKODUCED BY STEINHEIL. 



G. P. BOND, 



DIRECTOR OF THE OBSERVATORY OK HARVARD COLLEGE. 



[From the Proceedings of the American Academy of Arts and Sciences, Vol. VI.] 



PRINTED FROM THE STURGIS FUND FOR THE OBSERVATORY 
OF HARVARD COLLEGE. 






GAMBRIB GE: 
WELCH, BIGELOW, AND COMPANY, 

PRINTERS TO THE UNIVERSITY. 

1863. 



Svstein of Gauss. 



&<J 



System of Herschel. 




1 



.:- -- 



^l : \ 




Svsf eni of Franeahofer 





— 



. 



ON THE 



NEW FORM OE THE ACHROMATIC OBJECT-GLASS. 



In June, 1860, Professor Steinheil communicated to the Royal 
Academy of Sciences at Munich * a notice of an object-glass of thirty- 
six lines aperture, executed at his optical establishment according to 
the system of curves proposed by Gauss in an article published in the 
Zeitschrift fur Astronomie of Lindenau and Bohnenberger, in 1817.f 

This telescope, and subsequently another % of similar form, but 
larger, have been carefully tested, and, in the opinion of competent 
judges, they have exhibited a more complete achromatism, and in 
other respects more perfect definition, than was to be found with object- 
glasses of the ordinary form, of equaj dimensions. 

Some part of this superiority may be attributable to the manner of 
mounting the lenses, which admits of readily changing their relative 
positions so as to effect the best adjustment by actual trial ; a provision 
undoubtedly of considerable value, but perhaps equally applicable in 
the old system, if a slight separation of the inner surfaces of the 
crown and flint lenses were made one of the conditions for determining 
the curves. By this means, as Steinheil has remarked, we may not 
only diminish outstanding errors in the object-glass, but also, to. some 
extent, the aberrations of the eyepiece, and even defects in the eye 
itself. There seems, however, to be no reason to doubt that these 
object-glasses owe their excellence mainly to the improved theory of 
their curves. 

Among other advantages, the new combination admits of larger 
angles of aperture than would otherwise be practicable, without com- 
promising the clearness of the definition. It is here, in fact, that the 
value of the improvement is best illustrated. Any shortening of the 
focal length accomplished without sacrificing illuminating power, or 

* Sitzungsberichte der konigl. bayer. Akademie der Wissenschaften zu Miin- 
chen, 1860, II. 160. 

t Lindenau und Bohnenberger, Zeitschrift fur Astronomie, Nov., Dec. 1817, 
IV. 345. 

% Sitzungsberichte, 1860, V. 662. 



defining qualities, is a substantial gain in more than one direction. 
It reduces the telescope to a more manageable size, which, in one of 
the larger class, is a matter of the first importance, for not only is the 
size of the dome and building required to protect it, and, in general, 
the cost of all the accessory apparatus necessary for its efficiency, 
largely diminished by a reduction in the length of the focus, but the 
facility of using it depends also very much on the same condition. 
Again, by shortening the tube, we apply the best means of reducing 
its flexure, — one of the most intractable of all sources of error in 
meridian instruments. 

Hitherto, in the practice of the best opticians, the apertures of the 
largest object-glasses have not exceeded T ^ of the focal length, which 
is the proportion in Mr. Clark's 18.5-inch lens. With those of mod- 
erate size the ratio of T *g- to -^ has been successfully employed. Of 
such, the object-glasses of the vertical circle and prime vertical instru- 
ment at Poulkova, of 6-inch aperture, are examples of remarkable 
excellence. At present, however, Messrs. Merz are prepared to ex- 
tend the ratio of -^ even to lenses of 19£ inches (English) aperture ; 
a gain in the surface exposed to the light for the same focal length, of 
nearly seventy per cent. Steinheil has stated, that, with the Gaussian 
objectives, ratios of the aperture to the focal length as large as T \j- 
can be used for the largest refractors.* It must be remembered, that, 
owing to the strong curvature of the surfaces, the light has to traverse 
a greater thickness of the glass, and must experience more than ordi- 
nary loss from extinction. Perhaps, also, there will be a sensibly 
greater loss from reflection, from the greater inclination of the incident 
ray to the surface near the margin. The gain in area will, therefore, 
not represent precisely the increase in illuminating power. 

There are two other objections to the new construction which may 
be thought in some measure to counterbalance its special advantages : 
one of these is the much greater depth of its curves, suggesting, per- 
haps without sufficient foundation, practical difficulties of workmanship. 
That they have been actually overcome in lenses of moderate size is 
certainly the best reason for anticipating success when the trial is made 
on a larger scale. It is further evident, from the peculiar form of the 
lenses, each of which is a meniscus, that, if they are worked out of flat 
discs, as usual, greater thickness of material will be required. This 

* Sitzungsberichte der konigl. bayer. Akad. der Wiss., 1860, V. 663. 



would increase the difficulty, already so great, of procuring suitable 
glass. It is possible that the material could be accommodated nearly 
to the ultimate form of the lenses, just as, in the present process of 
manufacture, an irregular mass is moulded into a flat disc, approxi- 
mating to the shape required. It does not appear that either of these 
obstacles would long remain in the way of the general adoption of the 
new system, if its advantages were distinctly recognized, and sufficient 
inducements were offered to artists and to the manufacturers of optical 
glass to turn their efforts in this direction. 

The contrast presented in the character of the curves in the two 
combinations, which is so decided that the eye at once distinguishes 
between them without any occasion for measurement or exact com- 
parison, is very remarkable ; for if the superiority of Gauss's com- 
bination be admitted, it shows that the practice of opticians has been 
confined to a region altogether removed from that in which the best 
system is to be found. In this they have only adopted the recommen- 
dations of the many eminent mathematicians who have treated of the 
theory of the achromatic object-glass. 

The question proposed in this theory is to ascertain that form and 
disposition of the surfaces of two or more lenses, composed of materials 
of different dispersive powers, which shall most effectually destroy the 
aberrations of color and of figure. The problem, in the form in which 
it has been practically presented, is indeterminate, so that, for instance, 
in the case of lenses of crown and flint glass, " For every lens of 
crown-glass of positive focus, whatever the radii of its surfaces may 
be, a lens of flint-glass can be computed which will form, when united 
with it, an achromatic object-glass," — achromatic, that is to say, in the 
limited sense in which the term is commonly accepted. 

This allows, of course, of a great range in the choice of curves, and 
a variety of conditions have been proposed for determining the selec- 
tion. In one respect only has there been a general consent of authori- 
ties. The front lens has always been convex on both surfaces. But 
it would seem that in this particular the direction given to the investi- 
gation has not been fortunate. It is at least an oversight, that the 
relative importance of the two principal sources of indistinctness has 
not been kept prominently in view. For while it is admitted that the 
chromatic dispersion is the chief source of indistinctness, the arbitrary 
condition has not been determined with special reference to this cir- 
cumstance. 



6 

This omission has been supplied by Gauss, who has given attention 
mainly to the more complete elimination of the aberration of color, 
while, at the same time, his expectations that this could be done with, 
out sensibly increasing the spherical aberration, have been fully re- 
alized in the performance of the new object-glasses. Indeed, it de- 
serves notice that the resulting curves bear a considerable resemblance 
to one of the systems which has been designed with express reference 
to the correction of the spherical aberration. Allusion is here made 
to the forms deduced by Herschel * for the elimination of the spherical 
aberration of diverging, as well as parallel rays. From the compari- 
sons subjoined, it will be seen that one of the solutions satisfying 
his equations approximates nearly to Gauss's system, while the other 
approaches to a form employed by Frauenhofer. So far, therefore, 
as this holds good, each fulfils the conditions proposed in Herschel's 
theory. 

As Gauss has published neither the mathematical investigation of the 
subject, nor even the final equations from which his curves were com- 
puted, we have not the means of deciding with entire certainty, whether 
the resemblance referred to is merely accidental, or whether it expresses 
an affinity involved in the nature of the problem. But the latter seems 
the more probable explanation. The numerical values of the radii in 
his system, computed for a special case, are here transcribed from his 
original memoir, f after reducing them to a focal length, for the two 
lenses combined, of twenty-one French feet, for the sake of compari- 
son with the large Munich refractors. 

I. Gauss's Curves. 

ft. 
1st surface of the crown lens, convex, radius = -f- 2.535 
2d " " " concave, " = — 7.521 

1st " " flint lens, convex, " = + 3.123 

2d " " " concave, " = — 2.084 % 

Compound focus, = 21.00 

In No. 1289 of the Astronomische Naclirichten Oudemans has given 
the following measurements of an object-glass made by Frauenhofer 
for the Equatorial of the Observatory at Utrecht. The numbers have 
been reduced to the same unit as before, assuming the focal length 
from Astr. Nach. 1281. 

* Phil. Trans., 1821, p. 258. 
t Zeitschrift fur Astron., IV. 350. 

X This number has been corrected to accord with the erratum noticed at the end 
of the volume cited. 



II. Frauenhofer 's Curves. 

1st surface of the crown lens, convex, radius, = -|- 14.157 
2d " " " " " = + 5 - 635 

1st " " flint lens, concave, " = — 5.775 

2d " " " convex, " = + 25.945 

Compound focus, = 21.00 

Another of his object-glasses, probably computed from a similar for- 
mula, but for glass of slightly different refractive and dispersive powers, 
has values of the radii as follows * : — 

III. Frauenhofer 's Curves. 

1st surface of the crown lens, convex, radius, = -|- 15.430 
2d " " " " " = + 6 - 144 

1st " " flint lens, concave, " = — 6.262 

2d " " " convex, " = + 22.461 

Compound focus, = 21.00 

These numbers we will now compare with the two solutions of Her- 
schel's equations, using the notation I, r and r', to denote the reciprocals 
of the compound focal length and of the radii of the front surfaces of the 
two lenses. The substitution of the values of the indices of refrac- 
tion and of the dispersive powers which have been used by Gauss for 
computing the system I. gives the relations f : — 

== 2.3200 r 2 — 21.31 Ir + 59.57 I 2 -f 3.5792 lr> — 1.4233 r' 2 
= 6.6400 r — 24.95 I — 4.1119 r' 

From which we have 

== — 1.3917 r 2 + 12.37 Ir — 14.56 I 2 
with the roots 

j = 7.4922, and j = 1.3964, 

which afford the subjoined two sets of values. 
IV. HerscheVs Curves. 

j = 7.4922 

ft. 
1st surface of the crown lens, convex, radius, = -j- 2.803 
2d " " " concave, " =— 9.525 

1st " " flint lens, convex, " = -|- 3.482 

2d " " " concave, " = — 2.361 

Compound focus, " = 21.00 

* Zeitschrift fur Astron., IV. 352. 

t In the equation (z) Phil. Trans. 1821, p. 258, the coefficient of to 2 has been 

corrected from 2 " ~^ * to 3| ', t ~^~ l . Vide Article on Light, Encyc. Met., p. 424. 



LIBRARY OF CONGRESS 



003 608 306 2 



V. 4 = 1.3964 



ft. 
1st surface of the crown lens, convex, radius, =r= -f- 15.038 
2d " " " convex " = + 5.397 

1st " " flint lens, concave, - " = — 5.507 

2d " " " convex, " = -f- 22.135 

Compound focus, == 21.00 

With radii proportional to these numbers the figures in the accom- 
panying Plate have been constructed, representing sections of the dif- 
ferent object-glasses, each having a focal length of two feet, and an 
aperture of nearly four inches. The ratio of the aperture to the 
focal length has been taken larger than can be adopted in practice, 
in order to exaggerate the amount of curvature. It will be seen 
that the curves in the systems of Gauss and Frauenhofer may be 
nearly represented by the two solutions of Herschel's equations.* 

It follows that Gauss's form, originally designed to secure a more 
complete elimination of the chromatic dispersion, must be also rather fa- 
vorable than otherwise as regards the correction of the aberration of 
figure. It may be remarked, further, that his investigation, neglect- 
ing the thickness and distance of the lenses, leads to an equation of 
the fourth degree, which has no solution corresponding to V., nor to the 
above values of the radii used by Frauenhofer. On the other hand, if 
the curves in III. and V. have been derived from substantially the same 
theory, which seems a probable inference, it is scarcely possible that 
Frauenhofer should not have had at some time under consideration the 
system represented by the other solution of the equations, which would 
have conducted to forms approximating very nearly to the system of 
Gauss. 

* The refractive and dispersive powers in III., and probably in II., differ by small 
amounts from those used in computing IV. and V. ; moreover, in the latter, the effect 
of the thickness of the lenses and of their distance from each other has not been in- 
cluded, so that the numbers to be strictly comparable would require a small cor- 
rection. The values V., computed with the elements of refraction and disper- 
sion used for III., neglecting only the correction for thickness, become 

ft. 
+ 14.212 
+ 6.349 
— 6.488 
+ 25.375 
21.00 



q&n 




A 




003 



Hollinger Corp. 
pH 8.5 



